Generic mean of function derivative

Assume a discrete function $f(x)$, with mean $\mu$ and standard deviation $\sigma$. The mean and standard deviation of the function derivative $f^{\prime }(x)$ should be calculated in the following way:

$${\mu }^{\prime }=E[f^{\prime }(x)]=\frac{1}{N}\sum _i^N{f}^{\prime }(x_i)=\frac{1}{N}\sum _i^N{f}_i^{\prime }=\frac{1}{N}\sum _i^N\frac{\text{d}{f}_i}{\text{d}t}$$

Assume that $f$ is linear in x: $f_i=k{x}_i+m$, therefore $\frac{\text{d}{f}_i}{\text{d}t}=k\frac{\text{d}{x}_i}{\text{d}t}$, and hence:

$${\mu }^{\prime }=\frac{k}{N}\sum _i^N\frac{\text{d}{x}_i}{\text{d}t}=k{\mu }_{x^{\prime }}$$

Conclusion: Impossible to determine the mean of a function derivative based on the mean of the underived function.